# Answer a,b and c Transcribed Image Text: 2.Č Let V be an inner product space and v1, , Vn E V.

Excerpt |
---|

v;) O for 1 < i + j < n and (v;, v;) = i for 1 < i < n, then V1, V2, · · · , Vn are linearly independent. ... (f) If S is a subset of V, then (S) = S. |

Answer a,b and c Transcribed Image Text: 2.Č

Let V be an inner product space and v1,

, Vn E V. For each of

…

the following statements,

• write (T) if it is true and give a short proof, or

• write (F) if it is false and give a concrete counterexample.

(a) If every vector v E V is expressed as a linear combination of V1,·… , Vn

…

in a unique way, then dim V = n.

(b) If U and W are subspaces of V, then F = {u – w : u E U, w E W} is a

–

subspace of V.

(c) If U and W are subspaces of V such that u + w E U UW for all u E U

and w E W, then UUW = U or U U W = W.

(d) If U OW1 = UOW2 where U, W1, W2 are subspaces of V, then W1 = W2.

(e) If (vi, v;)

O for 1 < i + j < n and (v;, v;) = i for 1 < i < n, then
V1, V2, · · · , Vn are linearly independent.
...
(f) If S is a subset of V, then (S) = S.

### Need your ASSIGNMENT done? Use our paper writing service to score better and meet your deadline.

Click Here to Make an Order Click Here to Hire a Writer